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Integrate multiple time

hi everyone today we're going to talk about how to use integration by parts to find the antiderivative of a function to complete this problem we'll understand the integration by parts formula match the pieces of our function to the parts of our formula and then evaluate the integral let's take a look in this particular video we've been asked to evaluate the integral of e to the 7x times cosine of 2x DX now since this is an integration by parts problem the first thing that we need to do is identify in our function a value for u and a value for DV if we look at the integration by parts formula here what we can see is that it tells us is that if we're taking the integral of u times DV then we can set that equal to u times V minus the integral of V times D U so what we have to first identify is a value for u and a value for DV because the integral over here on the left hand side of our formula is representing the integral that we're taking here so we have to identify which part of this function here will be U and which part will be DV notice that there are no other parts of this integral so we have to assign this entire integral including the DX to U and DV somehow now for this particular problem I'll go ahead and say that whenever you have the exponential function whenever you have a here multiplied by a trigonometric function your best bet will be to assign you to the trigonometric function and DV to the exponential function so let's go ahead and do that we'll say that U is equal to cosine of 2x and that means since U is equal to cosine of 2x that DV here the other part of the integral has to include everything else in our integral which would be e to the 7x and DX so we'll say that DV is equal to e to the 7x DX one thing to note here is that DX will always be here with your DV so now we need to since we have U and DV we need to take the derivative of U to find D U and we need to take the integral of DV to find V so the next thing that we have to do now that we have U and DV is take the derivative of U to get D U and the integral or the antiderivative of DV to find V so taking the derivative of U we'll call it D you remember that the derivative of cosine is negative sine and we'll be using chain rule because cosine is our outside function and 2x is our inside function so we'll take the derivative of the outside function and get negative sine of 2x but then chain rule tells us that we have to multiply...